On actions of tori and quaternionic tori on products of spheres

Abstract: In this paper we study the actions of tori (standard compact tori, as well as their quaternionic analogues) on products of spheres. It is proved that the orbit space of a specific action of a torus on a product of spheres is homeomorphic to a sphere. A similar statement for a real torus $\mathbb{Z}_2^n$ was proved by the second author in 2019. We also provide a statement about arbitrary compact topological groups, generalizing the mentioned results, as well as the results of the first author about the actions of a compact torus of complexity one.